Class 12 Statistics Notes · GSEB

Real Line and Intervals

Real Line and Intervals — understand the real number line and read open, closed and half-open intervals with number-line diagrams. GSEB Class 12 Commerce Statistics notes with examples.

Last updated: 23 Sep 2026

Notes

The Real Line

Real Line

The real line is the number line on which every point corresponds to exactly one real number — integers, fractions, decimals and irrationals like √2 and π all live on it. It extends infinitely in both directions.

Every point is a real number

Your bank balance can be −₹500, ₹0 or ₹1,000.50 — every value sits on one line. That line is the real line.

Types of Intervals

An interval is a stretch of the number line. One question decides its type: do the endpoints belong? ( ) excludes them; [ ] includes them.

(a, b) = {x : a < x < b}

(a, b) — Both endpoints are excluded — x can get as close as you like to a or b, but never touch them.

Hollow ●

Endpoint excluded

Filled ●

Endpoint included

( )

Parentheses

Strict < or >

[ ]

Square brackets

Inclusive ≤ or ≥

Quick rule: ( ) = endpoint out, [ ] = endpoint in.

Interval Cheat Sheet

Type	Notation	Inequality	Endpoints
Open	(a, b)	a < x < b	Both excluded
Closed	[a, b]	a ≤ x ≤ b	Both included
Half-open	[a, b)	a ≤ x < b	a in, b out
Half-open	(a, b]	a < x ≤ b	a out, b in
Open (two-sided)	(−a, a)	−a < x < a	Both excluded
Closed (two-sided)	[−a, a]	−a ≤ x ≤ a	Both included

Modulus and Intervals

|x| = distance of x from 0. |x − a| = distance of x from a. So |x − a| < δjust means “x lies within δ of a” — an interval in disguise.

Modulus to Interval

|x| < a \iff -a < x < a

Modulus Explorer — |x − 2| < 5

Radius a = 5

Centre = 2

Modulus form

|x − 2| < 5

Inequality form

-3 < x < 7

Interval form

(-3, 7)

With ≤ the interval is closed [ ]; with < it is open ( ).

Solved Examples

Solved Example

Problem

Write the interval of all real numbers strictly between 2 and 5.

Solution

(2, 5), i.e. 2 < x < 5

Solved Example

Problem

Express |x − 3| {'<'} 2 as an interval.

Solution

(1, 5), i.e. 1 < x < 5

Solved Example

Problem

A canteen serves idli plates between ₹20 and ₹35, both included. Write this as an interval.

Solution

[20, 35], i.e. 20 ≤ x ≤ 35

Key Takeaways

The real line is the number line — every real number sits at one point on it.
(a, b) excludes endpoints; [a, b] includes them; half-open [a, b) mixes the two — read each bracket on its own.
|x − a| < δ is the open interval (a − δ, a + δ) — convert by expanding centre ± radius.
Exam rule: sign is < or > → use ( ); sign is ≤ or ≥ → use [ ].